The Symplectic Scheme For Rod Vibration Equation And The Conservative Scheme For Nonlinear Schr(?)dinger Equation

In practical application,we often need solve some differential equations withintrinsic properties such as conservation laws of energy,preserving symplecticityetc.In oder to get efficient and stable numerical methods of these equations,it isnecessary for us to construct the difference schemes that can keep the property.In thispaper,two kinds of differential equations with special properties, Four-oder RodVibration equation and Nonlinear Schr?dinger equation,are studied.We constructsymplectic schemes for Four-oder Rod Vibration equation and conservative schemefor the Nonlinear Schr?dinger equation and the coupled Nonlinear Schr?dingerequation.In the meantime,the stability and convergence of the schemes are analyzed. In the first chapter,the sympletic scheme of Four-oder Rod Vibration equationis derived.At first,the equation is transformed into a Hamilton system.Then two kindsof two-order and two-stage explicit symplectic scheme are constructed by PRKmethod and the stability of the corresponding schemes is analyzed.High-ordersymplectic scheme is derived by the constructed scheme and compositonmethod.Finally,the numerical example is given to verify the validity of the schemes. In the second and third chapter,the numerical solution method of nonlinearSchrodinger equation and coupled nonlinear Schr?dinger equation is studiedrespectively.We get conservative difference scheme by using difference method.Thetruncation error,stability and convergence of the schemes are analyzed.At last, thenumerical example is given to verify the validity of the schemes.